Aussie AI

Normalization Optimizations

  • Last Updated 25 April, 2026
  • by David Spuler, Ph.D.

Research has suggested various ways to speed up the normalization component. Examples of normalization improvements include:

  • Normalization alternatives
  • Normalization approximations
  • Removing normalization ("norm pruning")
  • Placement of normalization blocks (i.e. "pre-norm" vs "post-norm")
  • Fused normalization (see kernel operator fusion)

Normalization Implementation and Optimization

Normalization functions are not usually as significant as MatMul in terms of time cost, but they can still be significant. A typical normalization requires a multi-scan of all of the elements of the output vectors. And this is done multiple times per token throughout each inference phase.

Example: BatchNorm in C++: The batch normalization operation involves scanning the full vector, modifying each element so that it is re-centered to a zero mean, and re-scaled to a normal magnitude. A naive non-optimized version of C++ of BatchNorm looks like this: 

    void yapi_vector_batch_normalize_basic(    // Basic normalization (BatchNorm)
	float v[], int n, 
	float epsilon, // Smoothing term -- usually 1^e-5 (0.00005)
	float lambda, // Scaling term hyper-parameter (multiplication)
	float beta    // Bias/shift term hyper-parameter (addition)
    ) 
    {
	float fmean = yapi_vector_mean(v, n);  // Calculate "mean" (aka average)
	float variance = yapi_vector_variance_of_mean(v, n, fmean);  // Variance (sum-of-diffs-squared)
	
	float denom = sqrtf(variance + epsilon);  // like std. deviation, but smoothed by epsilon
	for (int i = 0; i < n; i++) {
		v[i] = (v[i] - fmean) / denom; // Normalize all elements to re-center and scale
	}
	yapi_vector_multiply_scalar(v, n, lambda);  // Scale all values by lambda hyper-param
	yapi_vector_add_scalar(v, n, beta);  // Add beta hyper-param to all values 
    }

This version is very inefficient with literally five scans of the entire vector. Loop fusion can obviously improve this, with the loops doing multiplication by lambda and the addition of beta merged into the prior for loop. Another optimization is to replace the division by "denom" with its reciprocal and a multiplication. Division is often an order-of-magnitude worse than multiplication.

Further optimizations become clear once we notice that each element of the vector has four operations being performed on it: subtracting the mean, dividing by the denominator, multiplying by lambda, and adding beta. We can use a loop fission optimization to split out the first two operations into separate loops, where simpler operations are probably faster with hardware acceleration. And then we notice that division and multiply are two versions of the same operation, so we can then use the loop fusion technique to merge the division-by-denom and multiply-by-lambda into a single multiplication by a combined scaling factor. Faster C++ code that has one less loop, and also calls atomic vector operations (easier to hardware accelerate), then results from these changes: 

	yapi_vector_add_scalar(v, n, -mean);  // Subtract the mean
	float scalef = lambda / denom;  // Combined scale factor
	yapi_vector_multiply_scalar(v, n, scalef);  // Scale by both denom and lambda 
	yapi_vector_add_scalar(v, n, beta);  // Add beta hyper-param to all values

Another way to optimize this code is simply to remove lambda and beta variables. Choosing lambda=1 and beta=0 means that the last two scalar multiplication and scalar addition loops can be avoided. However, there's now little benefit to removing lambda in the merged code above, although the add-beta loop can still be removed. Anyway, whether we can remove these parameters is not a speed decision, but depends on whether these two learned parameters are important to the overall model's capability. Note that there is also little value in trying to remove epsilon, as it is only used once in total.

Research on Optimizating Normalization: Research papers on fast versions of normalization functions:

Norm Optimizations: Book Excerpts and Blog Articles

Free online book excerpts with full text chapters online and free PDF downloads, and the Aussie AI blog, including related articles:

Approximating Normalization

Research on approximate normalization functions:

LayerNorm Approximation

Research papers on LayerNorm approximations:

Integer-Only Normalization

One approximations for normalization is to use integer-only arithmetic (see also overview of integers in inference). Research on integer-only normalization algorithms:

  • Y. Lin, Y. Li, T. Liu et al., “Towards fully 8-bit integer inference for the transformer model,” in Proc. of IJCAI, 2020, pp. 3759–3765. https://arxiv.org/abs/2009.08034 (Integers for weights, but also for Softmax, layer normalization, and other components, by replacing or approximating non-linear functions such as exponential and square-root.)
  • Zhewei Yao, Zhen Dong, Zhangcheng Zheng, Amir Gholami, Jiali Yu, Eric Tan, Leyuan Wang, Qijing Huang, Yida Wang, Michael Mahoney, Kurt Keutzer, HAWQ-V3: Dyadic Neural Network Quantization, Proceedings of the 38th International Conference on Machine Learning, PMLR 139:11875-11886, 2021, https://arxiv.org/abs/2011.10680 (Integer-only quantized weights and activations with INT4 or INT8, but also uses integers for batch normalization and residual connection components, too.)
  • A. Rock, A. Untether, O. Khalil, O. Shai, and P. Grouchy, 2022, INT8 Transformers for Inference Acceleration, 36th Conference on Neural Information Processing Systems (NeurIPS), PDF: https://neurips2022-enlsp.github.io/papers/paper_52.pdf
  • Y. Lin, Y. Li, T. Liu, T. Xiao, T. Liu, and J. Zhu, “Towards fully 8-bit integer inference for the transformer model,” the 29th International Joint Conference on Artificial Intelligence and the 17th Pacific Rim International Conference on Artificial Intelligence (IJCAI-PRICAI2020), 2020. https://arxiv.org/abs/2009.08034

Layer Normalization Placement Reordering (Pre-Norm/Post-Norm)

The original 2017 vanilla Transformer architecture (Vaswani et al, 2017) had a "post-norm" architecture. Subsequently, researchers found that switching to a "pre-norm" architecture, instead of post-norm, could fix one of the major problems with the original Transformer, namely that it was initially unstable in training, requiring a "warmup" phase. Pre-norm was found to stabilize early training and remove the need for any special handling in the warmup.

Since then, several researchers have explored where to place the layer normalization submodule. The general consensus seems to be that placing them before computations ("pre-norm") is better than after calcuations ("post-norm"). However, there are papers going either way, so there's still room for definitive research.

Research on Normalization Alternatives and Optimization

Research papers on different types of normalization or other alternatives:

General Research on Normalization

Research papers on normalization issues in general:

RMSNorm Research

RMSNorm is based on the Root Mean Squared (RMS) calculation. Research papers on RMSNorm include:

LayerNorm Optimizations

Research papers on optimizing LayerNorm:

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